Abstract
This paper briefly reviews some epistemological perspectives on the foundation of mathematical concepts and proofs. It provides examples of axioms and proofs, from Euclid to recent “concrete incompleteness” theorems. In reference to basic cognitive phenomena, the paper focuses on order and symmetries as core “construction principles” for mathematical knowledge. It then distinguishes between these principles and the “proof principles” of modern Mathematical Logic. It also emphasises the role of the blending of these different forms of founding principles for the purposes both of proving and of understanding and communicating the proof.
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Notes
- 1.
In my interpretation, existence, in the first axiom, is by construction and unicity by symmetry.
- 2.
Schrödinger stresses that a fundamental feature of Greek philosophy is the absence of ‘the unbearable division, which affected us for centuries… : the division between science and religion’ (quoted in Fraisopi 2009).
- 3.
Actually “signs” (σημεια, definition α): Boetius first used the word and the meaning of “point”. Note that a sign-point (σημειον) in Euclid is identified with the letter that names it (see Toth 2002).
- 4.
This section is partly borrowed from the introduction to Longo (2002).
- 5.
For more on the connections between “proof principles” and “construction principles” in mathematics and physics, see Bailly and Longo (2006).
- 6.
Concerning “concrete” incompleteness: An analysis of the nonprovability of normalisation for non-predicative Type Theory, Girard’s system F, in terms of prototype proofs is proposed in Longo (2002).
- 7.
- 8.
Masaccio and Piero invented the modern perspective, in Annunciations first (1400–1450), by the explicit use of points of converging parallel lines. As a matter of fact, the Annunciation is the locus of the encounter of the Infinity of God with the Madonna, a (finite) woman (see Panovsky 1991). Later, “infinity in painting”, by the work of Piero himself, became a general technique to describe finite spaces better.
- 9.
Klein and Clifford also stressed the role of symmetries in Euclidean Geometry: It is the only geometry which is closed under homotheties. That is, its group of automorphisms, and only its group, contains this form of symmetry.
- 10.
References
Bailly, F. & Longo, G. (2006). Mathématiques et sciences de la nature. La singularité physique du vivant. Paris: Hermann (Translation in English : Imperial College Press/ World Sci., London, 2010).
Bailly, F., & Longo, G. (2008). Phenomenology of incompleteness: From formal deductions to mathematics and physics. In Lupacchini (Ed.), Deduction, computation, experiment. Berlin: Springer.
Binney, J., Dowrick, N., Fisher, A., & Newman, M. (1992). The theory of critical phenomena: An introduction to the renormalization group. Oxford: Oxford University Press.
Boi, L. (1995). Le problème mathématique de l’espace. Berlin: Springer.
Boole, G. (1854). An investigation of the laws of thought. London: Macmillan.
Bottazzini, U. (2000). Poincaré, Pour la Science, N. Spécial, 4.
Bottazzini, U., & Tazzioli, R. (1995). Naturphilosophie and its role in Riemann’s mathematics. Revue d’Histoire des Mathématiques, 1, 3–38.
Brouwer, L. (1948). Consciousness, philosophy and mathematics. In Heyting (Ed.), Collected works (Vol. 1). Amsterdam: North-Holland.
Cantor, G. (1955). Contributions to the founding of the theory of transfinite numbers. Collected works. New York: Dove.
Cercignani, C. (1998). Ludwig Boltzmann: The Man Who trusted atoms. Oxford: Oxford University Press.
Connes, A. (1994). Non-commutative geometry. New York: Academic.
Dedekind, R. (1996). What are numbers and what should they be? In E. William (Ed.), From kant to hilbert: A source book in the foundations of mathematics (pp. 787–832). Oxford: Oxford University Press.
Dehaene, S. (1997). The number sense. Oxford: Oxford University Press. Longo’s review downloadable.
Einstein, A. (1915). Die feldgleichungen der gravitation (The field equations of gravitation) (pp. 844–847). Berlin: Königlich Preussische Akademie der Wissenshaften.
Fraisopi, F. (2009). Besinnung. Roma: Aracne.
Frege, G. (1884, [1980]). The foundations of arithmetic (Engl. transl. Evanston, London).
Friedman, H. (1997). Some historical perspectives on certain incompleteness phenomena. Retrieved May 21, 1997, 5 pp. Draft: http://www.math.ohio-state.edu/∼friedman/manuscripts.html
Gallier, J. (1991). What is so special about Kruskal’s theorem and the ordinal Γ0? Annals of Pure and Applied Logic, 53, 132–187.
Gauss, C. (1986). Disquisitiones Arithmeticae (1801) (A. A. Clarke, Trans.). New York: Springer.
Girard, J. (2001). Locus Solum. Mathematical Structures in Computer Science, 11(3), 323–542.
Girard, J., Lafont, Y., & Taylor, P. (1989). Proofs and types. Cambridge: Cambridge University Press.
Gödel, K. (1992). On formally undecidable propositions of principia mathematica and related systems (1931) (B. Meltzer, Trans.). New York: Dover.
Goldfarb, H., & Dreben, B. (Eds.). (1987). Jacques Herbrand: Logical writings. Boston: Harvard University Press.
Harrington, L., & Simpson, S. (Eds.). (1985). H. Friedman’s research on the foundations of mathematics. Amsterdam: North-Holland.
Heath, T. (1908). The thirteen books of Euclid’s elements (Heath’s translation and comments). Cambridge: Cambridge University Press.
Hilbert, D. (1899). Grundlagen der Geometrie Leipzig: Teubner (Translated by L. Unger as Foundations of Geometry, Open Court, La Salle, 1971).
Husserl, E. (1933). The Origin of Geometry (Appendix III of Krysis) (trad. fran. by J. Derida, Paris: PUF, 1962).
Kennedy, H. C. (2006). Life and works of Giuseppe Peano. Dordrecht, Holland: D. Reidel Publishing Company.
Longo, G. (2002). Reflections on incompleteness, or on the proofs of some formally unprovable propositions and prototype proofs in type theory. Invited lecture. In: P. Callaghan (Ed.), Types for Proofs and Programs, Durham, (GB), December 2000, Lecture Notes in Computer Science 2277. Berlin: Springer.
Longo, G. (2005). The cognitive foundations of mathematics: Human gestures in proofs and mathematical incompleteness of formalisms. In M. Okada & P. Grialou (Eds.), Images and reasoning (pp. 105–134). Tokyo: Keio University Press.
Longo, G. (2006). Sur l’importance des résultats négatifs. Intellectica, 40(1), 28–43. see http://www.di.ens.fr/users/longo/ for a translation in English.
Longo, G. (2009). Critique of computational reason in the natural sciences. In E. Gelenbe & J.-P. Kahane (Eds.), Fundamental concepts in computer science (pp. 43–70). London: Imperial College Press/World Scientific.
Longo, G. (2011). Interfaces of incompleteness. Italian version in La Matematica (Vol. 4), Einuadi.
Longo, G. & Viarouge, A. (2010). Mathematical intuition and the cognitive roots of mathematical concepts. In L. Horsten & I. Starikova (Eds.), Invited paper, Topoi, Special issue on Mathematical Knowledge: Intuition, Visualization, and Understanding, 29(1), 15–27.
Padua, A. (1967). Logical introduction to any deductive theory. In J. van Heijenoort (Ed.), A source book in mathematical logic (pp. 118–123). Boston: Harvard University Press.
Panovsky, E. (1991). Perspective as symbolic form. New York: Zone Books.
Peano, G. (1967). The principles of arithmetic presented by a new method. In J. van Heijenoort (Ed.), A source book in mathematical logic (pp. 83–97). Boston: Harvard University Press.
Petitot, J. (2003). The neurogeometry of pinwheels as a sub-Riemannian contact structure. The Journal of Physiology, 97(2–3), 265–309.
Poincaré, H. (1908). Science et Méthode. Paris: Flammarion.
Rathjen, M., & Weiermann, A. (1993). Proof-theoretic investigations on Kruskal’s theorem. Annals of Pure and Applied Logic, 60, 49–88.
Riemann, B. (1854, [1873]). On the hypothesis which lie at the basis of geometry (English trans. by W. Clifford). Nature 8(1873), 14–17, 36–37.
Tappenden, J. (1995). Geometry and generality in Frege’s philosophy of arithmetic. Synthese, 102(3), 25–64.
Toth, I. (2002). Aristotele ed i fondamenti assiomatici della geometria. Milano: Vita e Pensiero.
Turing A. M. (1950) Computing machines and intelligence, Mind, LIX. Reprinted in M. Boden (Ed.), Oxford University Press, 1990.
Turing, A. M. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society, B237, 37–72.
van Fraassen, B. (1993). Laws and symmetry. Oxford: Oxford University Press.
Vasari, A. (1998). The lives of the artists. Oxford: Oxford University Press.
Weyl, H. (1949). Philosophy of mathematics and of natural sciences. Princeton: Princeton University Press.
Weyl, H. (1952). Symmetry. Princeton: Princeton University Press.
Zellini, P. (2005). A brief history of infinity. New York: Penguin.
Acknowledgements
Many discussions with Imre Toth, and his comments, helped to set my (apparently new) understanding of Euclid on more sound historical underpinnings. Rossella Fabbrichesi helped me to understand Greek thought and the philosophical sense of my perspective. My daughter Sara taught me about “infinity in the painting” in the Italian Quattrocento. The editors proposed a very close revision for English and style.
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Longo, G. (2012). Theorems as Constructive Visions. In: Hanna, G., de Villiers, M. (eds) Proof and Proving in Mathematics Education. New ICMI Study Series, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_3
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